For a positive integer $n$, $r(n)$ denote the sum of the remainders when $n$ is divided by $1, 2,..., n$ respectively. Prove that $r(k) = r(k -1)$ for infinitely many positive integers $k$.

Let $\sigma(n)$ denote the sum of positive divisors of a number $n$. A positive integer $N=2^r b$ is given, where $r$ and $b$ are positive integers and $b$ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and $\sigma(b)$ are coprime. (J. Antalan, J. Dris)

Find the largest positive integer that cannot be written in the form $a + bc$ for some positive integers $a, b, c$, satisfying $a < b < c$.

Let $ \mathcal G$ be the set of all finite groups with at least two elements. a) Prove that if $ G\in \mathcal G$, then the number of morphisms $ f: G\to G$ is at most $ \sqrt [p]{n^n}$, where $ p$ is the largest prime divisor of $ n$, and $ n$ is the number of elements in $ G$. b) Find all the groups in $ \mathcal G$ for which the inequality at point a) is an equality.

Find the sum of the four smallest prime divisors of $2016^{239} - 1$.

Let $ABC$ be a triangle where $AB > BC$, and $D$ and $E$ be points on sides $AB$ and $AC$ respectively, such that $DE$ and $AC$ are parallel. Consider the circumscribed circumference of triangle $ABC$. A circumference that passes through points $D$ and $E$ is tangent to the arc $AC$ that does not contain $B$ at point $P$. Let $Q$ be the reflection of point $P$ with respect to the perpendicular bisector of $AC$. The segments $BQ$ and $DE$ intersect at $X$. Prove that $AX = XC$.

Positive numbers $a$ and $b$ satisfy $a^{1999}+b^{2000}{\ge}a^{2000}+b^{2001}$.Prove that $a^{2000}+b^{2000}{\leq}2$. _____________________________________ Azerbaijan Land of the Fire :lol:

Let $N$ be the number of functions $f$ from $\{1,2,\ldots, 8\}$ to $\{1,2,3,\ldots, 255\}$ with the property that: [list] [*] $f(k)=1$ for some $k \in \{1,2,3,4,5,6,7,8\}$ [*] If $f(a) =f(b)$, then $a=b$. [*] For all $n \in \{1,2,3,4,5,6,7,8\}$, if $f(n) \neq 1$, then $f(k)+1>\frac{f(n)}{2} \geq f(k)$ for some $k \in \{1,2,\ldots, 7,8\}$. [*] For all $k,n \in \{1,2,3,4,5,6,7,8\}$, if $f(n)=2f(k)+1$, then $k<n$. [/list] Compute the number of positive integer divisors of $N$. [i]2021 CCA Math Bonanza Individual Round #15[/i]

For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?

A number $N$ is defined as follows: \[N=2+22+202+2002+20002+\cdots+2\overbrace{00\ldots000}^{19~0\text{'s}}2\] When the value of $N$ is simplified, what is the sum of its digits? $\textbf{(A) }42\qquad\textbf{(B) }44\qquad\textbf{(C) }46\qquad\textbf{(D) }50\qquad\textbf{(E) }52$

Assume natural number $n\ge2$. Amin and Ali take turns playing the following game: In each step, the player whose turn has come chooses index $i$ from the set $\{0,1,\cdots,n\}$, such that none of the two players had chosen this index in the previous turns; also this player in this turn chooses nonzero rational number $a_i$ too. Ali performs the first turn. The game ends when all the indices $i\in\{0,1,\cdots,n\}$ were chosen. In the end, from the chosen numbers the following polynomial is built: $$P(x)=a_nx^n+\cdots+a_1x+a_0$$ Ali's goal is that the preceding polynomial has a rational root and Amin's goal is that to prevent this matter. Find all $n\ge2$ such that Ali can play in a way to be sure independent of how Amin plays achieves his goal.

In triangle $ABC$, the interior and exterior angle bisectors of $ \angle BAC$ intersect the line $BC$ in $D $ and $E$, respectively. Let $F$ be the second point of intersection of the line $AD$ with the circumcircle of the triangle $ ABC$. Let $O$ be the circumcentre of the triangle $ ABC $and let $D'$ be the reflection of $D$ in $O$. Prove that $ \angle D'FE =90.$

The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$, compute $P(1)$.

The sequence of functions $f_n:[0,1]\to\mathbb R$ $(n\ge2)$ is given by $f_n=1+x^{n^2-1}+x^{n^2+2n}$. Let $S_n$ denote the area of the figure bounded by the graph of the function $f_n$ and the lines $x=0$, $x=1$, and $y=0$. Compute $$\lim_{n\to\infty}\left(\frac{\sqrt{S_1}+\sqrt{S_2}+\ldots+\sqrt{S_n}}n\right)^n.$$

Prove that for every integer $S\ge100$ there exists an integer $P$ for which the following story could hold true: The mathematician asks the shop owner: ``How much are the table, the cabinet and the bookshelf?'' The shop owner replies: ``Each item costs a positive integer amount of Euros. The table is more expensive than the cabinet, and the cabinet is more expensive than the bookshelf. The sum of the three prices is $S$ and their product is $P$.'' The mathematician thinks and complains: ``This is not enough information to determine the three prices!'' (Proposed by Gerhard Woeginger, Austria)

There are $2022$ students in winter school. Two arbitrary students are friend or enemy each other. Each turn, we choose a student $S$, make friends of $S$ enemies, and make enemies of $S$ friends. This continues until it satisfies the final condition. [b]Final Condition[/b] : For any partition of students into two non-empty groups $A$, $B$, there exist two students $a$, $b$ such that $a\in A$, $b\in B$, and $a$, $b$ are friend each other. Determine the minimum value of $n$ such that regardless of the initial condition, we can satisfy the final condition with no more than $n$ turns.

An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order. [asy] size(3cm); pair A=(0,0),D=(1,0),B,C,E,F,G,H,I; G=rotate(60,A)*D; B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A; draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]

Establish if there exists a finite set $M$ of points in space, not all situated in the same plane, so that for any straight line $d$ which contains at least two points from M there exists another straight line $d'$, parallel with $d,$ but distinct from $d$, which also contains at least two points from $M$.

The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$. [i]Proposed By Pedram Safaei[/i]

Prove that there exists an infinity of irrational numbers $x,y$ such that the number $x+y=xy$ is a nonnegative integer.

Let $k > 2$ be an integer. A positive integer $l$ is said to be $k-pable$ if the numbers $1, 3, 5, . . . , 2k - 1$ can be partitioned into two subsets $A$ and $B$ in such a way that the sum of the elements of $A$ is exactly $l$ times as large as the sum of the elements of $B$. Show that the smallest $k-pable$ integer is coprime to $k$.

The quadrilateral $ABCD$ is an isosceles trapezoid with $AB = CD = 1$, $BC = 2$, and $DA = 1+ \sqrt{3}$. What is the measure of $\angle ACD$ in degrees?

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

Given a tetrahedron $ABCD$. Let $x = AB \cdot CD, y = AC \cdot BD$ and $z = AD\cdot BC$. Prove that there exists a triangle with the side lengths $x, y$ and $z$.

Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.) For which $n$ is it possible for Mark to achieve this? [i]Proposed by Michael Ren[/i]